impact of cross aisles in a rectangular warehouse: a...

Chapter 5

Impact of Cross Aislesin a RectangularWarehouse:A Computational Study

Gurdal Ertek, Bilge Incel, and Mehmet Can Arslan

CONTENTS

5.1 Introduction ..................................................................................................... 985.2 Related Literature ........................................................................................... 1035.3 Vaughan and Petersen Model........................................................................ 1045.4 Modified Model.............................................................................................. 1075.5 Algorithms to Identify Best Storage-Block Lengths Li .................................. 1085.6 Experimental Design ..................................................................................... 1095.7 Analysis of Experimental Results .................................................................. 110

5.7.1 Savings in Case 2 Compared to Case 1 ............................................. 115

5.7.2 Impact of T, M, and D in Case 1........................................................ 117

5.7.3 Best Number of Cross Aisles in Case 1 ............................................. 1185.8 Conclusions and Future Work....................................................................... 120

Lahmar / Facility Logistics AU8518_C005 Page Proof page 97 11.8.2007 1:50pm Compositor Name: VAmoudavally

97

5.9 Acknowledgments ......................................................................................... 121References............................................................................................................... 121Appendix ................................................................................................................ 123

Abstract Order picking is typically the most costly operation in a ware-house, and traveling is typically the most time-consuming task within orderpicking. In this study, we focus on the layout design for a rectangular ware-house, a warehouse with parallel storage blocks with main aisles separatingthem.We specifically analyze the impact of adding cross aisles that cut storageblocks perpendicularly,which can reduce travel times during order picking byintroducing flexibility in going from one main aisle to the next. We considertwo types of cross aisles, those that are equally spaced (Case 1) and those thatare unequally spaced (Case 2), which respectively have equal and unequaldistances among them. For Case 2, we extend an earlier model and present aheuristic algorithm for finding the best distances among cross aisles. We carryout extensive computational experiments for a variety of warehouse designs.Our findings suggest that warehouse planners can obtain great travel-time savings by establishing equally spaced cross aisles, but little additionalsavings in unequally spaced cross aisles. We present a look-up table thatprovides the best number of equally spaced cross aisles when the numberof cross aisles (N) and the length of the warehouse (T) are given. Finally,when the values of N and T are not known, we suggest establishing threecross aisles in a warehouse.

5.1 Introduction

Order picking is generally the most significant operation in a warehouse,accounting for approximately 60 percent of all operational costs in a typicalwarehouse (Frazelle, 2001). Cost of order picking is affected by thedecisions regarding the facility layout and the selection of storage andretrieval systems, and by the implemented strategies such as zoning, batch-ing, and routing. Travel cost is typically the largest cost component withinorder picking, accounting for about 50 percent of the costs associated withorder-picking activities (Frazelle and Apple, 1994 AQ1). Because order picking,specifically traveling, is costly, reducing the travel time spent for orderpicking can significantly reduce operational costs.

In this chapter, we present the findings of a study that focuses on thestrategic layout decisions of how many cross aisles to establish within arectangular warehouse and how to determine the distances among them.A ‘‘rectangular warehouse’’ can be defined as a warehouse with equi-length


98 & Facility Logistics

parallel ‘‘storage blocks,’’ separated by aisles in between (see Figure 5.1).A subregion of a large warehouse, where routing decisions are made inde-pendently from the remaining regions, can also be considered as a rectangu-lar warehouse, given that it satisfies the structural properties describedearlier. A rectangular warehousemay have only ‘‘main aisles’’ which separatethe storage blocks vertically (Figure 5.1), or may also contain one or more‘‘cross aisles’’ perpendicular to the main aisles, which divide the storageblocks horizontally (see Figures 5.2 through 5.4). The main advantage ofcross aisles is that they enable savings in travel times, especially during theorder-picking operations. Figures 5.1 and 5.2 illustrate an example of arectangular warehouse where the creation of a cross aisle can reduce thetravel distance while picking an order with seven items: The addition of thecross aisle (Figure 5.2) shortens the travel distance by enabling shortcutsfrom the fifth main aisle to the fourth main aisle and from the fourth mainaisle to the third main aisle.

Storage blocks typically consist of steel racks that are installed on thewarehouse floor permanently during the construction of a warehouse.Thus, the decisions regarding the quantities and dimensions of storage

Case 0: No cross aisles (N = 0)

Mai

n ai

sle

5

Mai

n ai

sle

4

Mai

n ai

sle

3

Mai

n ai

sle

2

Mai

n ai

sle

1Start Finish

Figure 5.1 Case 0: A rectangular warehouse with four storage blocks, five mainaisles, and no cross aisles. (The vectors show a route to pick an order with sevenitems.)


Impact of Cross Aisles in a Rectangular Warehouse & 99

Mai

n ai

sle

5

Mai

n ai

sle

4

Mai

n ai

sle

3

Mai

n ai

sle

2

Mai

n ai

sle

1

Start

Cross aisle N

Finish

Figure 5.2 The warehouse displayed in Case 0 with an interior cross aisle with short-cuts linking main aisle 5 to main aisle 4 and main aisle 4 to main aisle 3.

Case 1: Equally spaced cross aisles

Mai

n ai

sle

M

Mai

n ai

sle

(M–1

)

Mai

n ai

sle

3

Mai

n ai

sle

2

Mai

n ai

sle

1

Start Finish

B W

Cross aisle N + 1

Cross aisle N

Cross aisle 1

Cross aisle 0

L

A

Figure 5.3 A rectangular warehouse with equally spaced cross aisles (Case 1).



blocks, main aisles, and cross aisles are strategic decisions. These decisionsshould be made considering many factors, including:

& The physical dimensions of the building& The characteristics of the materials to be stored, including physical

dimensions, weights, shelf lives, pallet sizes, and projected demandpatterns

& The characteristics of warehouse equipment such as forklifts andautomatic guided vehicles (AGVs)

& The quantity and capabilities of the workforce& The capabilities of the information system, that is, the warehouse

management system (WMS)

One classic challenge raised by these factors is how to incorporate theinteractions between different decision levels in the design and operationof warehouses (Rouwenhorst et al., 2000). For example, the strategicdecision of determining the best warehouse layout, the tactical decisionof assigning the products to the storage locations in the best way, and theoperational decision of determining the best order-picking routes are allinterdependent. In our study, we assume that the widths of the storageblocks, the main aisles, and the cross aisles are fixed, and that the items

Case 2: Unequally spaced cross aisles

Cross aisle N +1

Cross aisle N

Cross aisle 1

Cross aisle 0

Start Finish

L1

A

L2

LN+1

M 3 2M–1

Figure 5.4 A rectangular warehouse with unequally spaced cross aisles (Case 2).



stored in the warehouse all have the same demand frequencies. Evenunder these simplifying assumptions, the strategic decisions regardingthe number of cross aisles and the distances between them (the lengthsof storage blocks) have to be made by estimating the average traveldistance in order picking under a specific routing algorithm. Hence, inour study, we assume that the routing algorithm and the storage locationsof products are predetermined, and focus on the strategic decisionsregarding cross aisles.

For a rectangular warehouse, one can identify the following three caseswith respect to the number of cross aisles N and the distances in betweenthem:

Case 0: Warehouse with no cross aisles (N ¼ 0), as shown in Figure 5.1.Case 1: Warehouse with N equally spaced cross aisles (N� 1), as

shown in Figure 5.3.Case 2: Warehouse with N unequally spaced cross aisles (N� 1),

as shown in Figure 5.4.

In our study we seek answers to the following research questions regardingthe rectangular warehouse:

Should the cross aisles be established equally spaced (Case 1) orunequally spaced (Case 2)? In other words, should the storage blockshave an equal length or variable lengths? How much travel-time savingsdo cross aisles bring? Under which settings do cross aisles bring themost travel-time savings? How many cross aisles should there ‘‘ideally’’be in a rectangular warehouse? In other words, what is the best number ofcross aisles?

To answer these questions, we carry out extensive computationalexperiments reflecting a variety of warehouse settings with different valuesfor warehouse lengths (T), number of cross aisles (M), and pick densities(D). Based on a thorough analysis of our experimental results, we come upwith answers to the aforementioned research questions.

One unique aspect of our research is that we extensively apply thestarfield visualization technique from the field of information visualiza-tion. In ‘‘starfield visualization,’’ various fields of a dataset are mappedto the axes of a colored 2-D or 3-D scatter plot, and to the attributes ofthe glyphs (data points) such as color, size, and shape. ‘‘Informationvisualization’’ is the growing field of computer science that combinesthe fields of data mining, computer graphics, and exploratory data analy-sis (in statistics) in pursuit of visually understanding data (Spence, 2001;Keim, 2002). The ultimate goal in information visualization is to discoverhidden patterns and gain actionable insights through a variety of—possibly interactive—visualizations.



The use of a visualization approach in the analysis of our numericalresults will enable us to make important observations and develop mana-gerial insights. To the best of our knowledge, this is the first attempt wheredata/information visualization techniques are employed to this extent in thewarehousing and facility logistics literature.

5.2 Related Literature

Rouwenhorst et al. (2000) present a reference framework and classificationof warehouse design and operating problems. Van den Berg and Zijm(1999) provide another review of the warehousing literature that classifieswarehouse management problems. Sharp (2000) summarizes functionalwarehouse operations; database considerations; and tactical, strategic andoperational issues in warehouse planning and design.

Within the vast facility logistics literature, there exist studies that solelyfocus on order-picking routing and order batching for the purpose ofreducing travel time. An early study by Ratliff and Rosenthal (1983) solvesthe routing problem in order picking. Based on the number of aisles, theauthors propose an algorithm that solves the problem to optimality. Theystate that the algorithm computation time grows linearly in the number ofaisles, and is thus scalable for solving real-world problems.

Roodbergen and De Koster (2001) AQ2analyze the relationship betweenwarehouse layout and average travel time. They consider a rectangularwarehouse in which a single cross aisle divides the warehouse into twoequal-length blocks. The authors present a dynamic programming algorithmto determine the shortest order-picking routes, and show that the additionof the cross aisle decreases average order-picking time significantly.

In another study, Roodbergen and De Koster (2001) AQ3compare severalalgorithms for routing order pickers in a warehouse with more than onecross aisle. They introduce two new heuristics, combined and combinedþ,and compare them with the S-Shape, Largest Gap, and Aisle-by-Aisle heuri-stics in the literature. The authors prove through computational tests thatthe combinedþ heuristic performs best among the five heuristics. A branch-and-bound algorithm is used as a benchmark to compare the performancesof the generated heuristics.

De Koster et al. (1999) report a real-world application, where theysignificantly improve the efficiency of manual order-picking activities at alarge retail distribution center in the Netherlands. In the first stage of theirstudy, the authors apply a routing heuristic, which ensures that orderpickers pick items from both sides of an aisle. This heuristic alone achievesa 30 percent reduction in travel time, and consequently a saving of 1.2 orderpickers. In the latter stage of their study, the authors apply order batching,



time-savings method and a combined routing heuristic (De Koster andVan der Poort, 1998) jointly, and achieve 68 percent reduction in traveldistance and a saving of 3 to 4 pickers. This study is the perfect example ofhow the order-picking strategies and routing algorithms proposed in theliterature can be applied in the real world to achieve substantial savings.

Our study is mainly related to the work of Vaughan and Petersen (1999),who consider both layout and routing. Vaughan and Petersen are motivatedby the fact that cross aisles can reduce travel distances due to their flexibilityin order picking. The authors develop a shortest path pick sequencingmodel that is applicable to any number of equally spaced cross aisles(equal-length storage blocks) in the warehouse. Their model assumes thatall the items along an aisle are picked before proceeding to the next aisle,and the order picking progresses from the leftmost aisle to the rightmostaisle. This policy is referred to as ‘‘aisle-by-aisle policy.’’ The authors com-pute the optimal routes for a large number of randomly generated pickingrequests, over a variety of warehouse layouts and order picking parameters.Their results suggest that when the main storage-aisle length (T) is small, anexcessive number of cross aisles can increase the average travel distance.This is true especially when the number of storage aisles (M) is small, andwhen pick density is very small or very large. The authors warn that thesavings due to cross aisles diminish, even turn into losses, if the number ofcross aisles becomes excessive. This is because the extra distance to tra-verse the cross aisles increases the travel distances. Additionally, the authorsfind out that as the main storage-aisle length (T) increases the optimalnumber of cross aisles also increases and report that cross aisles are mostbeneficial for longer warehouses.

5.3 Vaughan and Petersen Model

Vaughan and Petersen (1999) assume certain characteristics with respect tothe rectangular warehouses and order-picking policies. Because our studyis built on their model, which we will refer to as the V&P model, thefollowing assumptions are also valid for our model.

& There are parallel main aisles, and products are stored on both sidesof the main aisles.

& Each order includes a number of items to be picked, which aregenerally located in various main aisles.

& All the stocks of a particular item are stored in a single location.& Order pickers can traverse the aisles in both directions and change

directions within the main aisles.



& The main aisles are narrow enough to pick from both sides of theaisle without changing position.

& The main aisles are wide enough such that two or more orderpickers can operate in the main aisle at the same time.

& There are two natural cross aisles in the warehouse, at the head andrear of the warehouse.

& Cross aisles are not used to store items; they are only used to pass tothe next main aisle.

& The items of an order are collected in a single tour.

Block lengths are determined by the locations of the cross aisles that dividemain aisles perpendicularly. In our study, the ‘‘number of cross aisles’’refers to the number of interior cross aisles, which are between the defaulthead and rear cross aisles. We assume that picking routes start and end atthe southeast and southwest corners of the warehouse, respectively. Eventhough some research assumes that order picking ends at the starting point(De Koster and Van der Poort, 1998; Roodbergen and De Koster, 2001 AQ3), thisdoes not make a great change in travel distance (and thus travel time).Petersen (1997) notes that this change results in at most 1 percent deviationin travel distance.

The dynamic programming algorithm developed by Vaughan and Peter-sen (1999) finds the optimal route to pick an order under the aisle-by-aislepolicy. The complete notation for their so called shortest path model is asfollows:

L: Length of a storage blockT: Length of the warehouse (equal to the length of main aisles), T¼

(N þ 1)LM: Number of main aislesN: Number of interior cross aisles (The total number of cross aisles is

N þ 2)A: Width of a cross aisle. (This parameter is essential for the calcula-

tion of the best aisle-by-aisle route. The model assumes that anorder picker walks along the center of the cross aisles.)

This walking pattern is illustrated in Figure 5.5 and the additional distanceof A/2 to walk to the middle of the cross aisle is reflected in the formulas forB1m and B2m.

B: Width of a main aisleC: Width of a storage block



The notation until now is related to the warehouse layout. The notationbelow is given for a particular order to be picked:

Km: The number of items to be picked by the order picker from

main aisle m¼ 1, 2, . . . , M. Thus, the order consists ofPMm¼1

Km

items in totalXm(t): The location of an item t in main aislem¼ 1, 2, . . . ,M, and t¼ 1,

2, . . . , Km (undefined if Km¼ 0) where 0 � Xm(t) � T

(The expressions listed below are demonstrated in Figure 5.5a.)

Xmþ: The location of the item at the south-most location (highest

value) in main aisle m (undefined if Km¼ 0), i.e., Xmþ ¼

maxt

fXmðtÞgXm

�: The location of the item at the north-most location (smallestvalue) in main aisle m (undefined if Km¼ 0), i.e.,Xm

� ¼ mint

fXmðtÞgCm(i,j): The total vertical travel distance required to pick all the items

in main aisle m, if main aisle m is entered at cross aisle i andexited to main aisle m-1 at cross aisle j

B1m(i,j): The length of forward-tracking leg required to pick the itemsin main aisle m to the north of cross aisle h, h¼min (i,j)

(a)

L

i

jj

1Entrance

crossaisle

LB1m(i,j)

Cm(i,j)

B2m(i,j)

L2 L

L3 L

L4 L

Xm−

Xm+

(b)

L1

i

1Entrance

crossaisle

L1 B1m(i,j)

Cm(i,j)

B2m(i,j)

L22 L

2

L33 L

3

L44 L

4

Xm−

Xm+

Figure 5.5 (a) A warehouse with storage blocks of equal lengths and (b) awarehouse with storage blocks with unequal lengths. (Arrows represent the totalvertical travel distance to pick items in main aisle m when the main aisle isentered from the ith cross aisle and left from the jth cross aisle.)



B2m(i,j): The length of back-tracking leg required to pick the items inmain aisle m to the south of cross aisle h, h¼max (i,j)

fm(i): The minimum total picking distance required to pick all theitems in aisle m, m-1, m-2, . . . , 2, 1 if main aisle m is enteredat cross aisle position i

In the V&P model Cm, B1m, and B2m are calculated as follows:

Cm(i, j) ¼ B1m(i, j)þ i � jj(Lþ A)þ B2m(i, j)j ; where

B1m(i, j) ¼0 for Km ¼ 0,0 for X�

m � min (iL, jL),2½min (iL, jL)� X�

m for X�m < min (iL, jL);

þA(0:5þ ((min (iL, jL)� X�m)=L))�

8>><>>:

and

B2m(i, j) ¼0 for Km ¼ 0,0 for Xþ

m < max (iL, jL),2�max (iL, jL)� Xþ

m for Xþm � max (iL, jL):

þA(0:5þ ((Xþm �max (iL, jL))=L))

�8>><>>:

The dynamic programming equations for each stage are given as follows:

fm(i) ¼ minj

Cm(i, j)þ fm�1( j)f g, where f1(i) ¼ C1(i,N þ 1):

Stages of the dynamic programming are related to the main aisle numbers inthe warehouse. The desired shortest-path picking route is determined byevaluating fM(N þ 1).

5.4 Modified Model

Now we present our model that allows us to find the best routes accordingto the aisle-by-aisle heuristic for the case of unequally-spaced cross aisles(Case 2). The primary difference between our model and the V&P model isthat the storage blocks now have variable lengths Li (Figure 5.5b) instead ofa fixed length of L (Figure 5.5a) where Li is the length of the ith storage blockfor i¼ 1, . . . , N þ 1. Thus, the length of the warehouse T which is equal to

the length of the main aisles can be expressed as T ¼ PNþ1

i¼1Li.



Next, we define two new notations that give us the indices of theblocks where the north-most and south-most items within an aisle arelocated:

Blockof(Xmþ): Index of storage block Li in main aisle m where Xm

þ islocated for i¼ 1, 2, . . . , N þ 1.

Blockof(Xm�): Index of storage block Li in main aisle m where Xm

� islocated for i¼ 1, 2, . . . , N þ 1.

Finally, the Cm, B1m, and B2m values are calculated based on the modifieddefinitions of block lengths Li and the warehouse length T which can beexpressed as:

Cm(i, j) ¼ B1m(i, j)þXmax (i, j)

s¼min (i, j)þ1

Ls þ i � jj jAþ B2m(i, j),

where

B1m(i, j)¼2 minXis¼1

Ls,Xjf¼1

Lf

!�X�

mþA(0:5þmin(i, j)�Blockof (X�m))

" #, and

B2m(i, j)¼ 2 Xþm�max

Xis¼1

Ls,Xjf¼1

Lf

!þA(0:5þBlockof (Xþ

m )�1�max(i, j))

#"

5.5 Algorithms to Identify Best Storage-BlockLengths Li

Given T, M, N, A, B, C, and D values, the problem of finding the beststorage-block lengths Li is a difficult problem. This is because the length of atour is found by solving a dynamic programming problem and the locationsof the items are uniformly distributed. The objective function to be minim-ized is the average travel distance (and thus, the average travel time) over allorders, with the optimal travel distance for each order computed throughdynamic programming optimization. We, thus, develop and implement twoheuristic search algorithms, namely GSA (grid search algorithm) and RGSA(refined grid search algorithm), to find the best Li values. GSA takes awarehouse (with its T, M, N, A, B, C values), a set of generated orders,and the number of grids as parameters, and identifies an initial solution,which consists of Li values. RGSA takes the solution of GSA as the initial



solution and carries out a search to reduce the average travel distance (i.e.,average travel time). These algorithms are given in pseudo-code and areexplained in the Appendix.

5.6 Experimental Design

The different values of model parameters that we have used in our compu-tational experiments are depicted in Table 5.1. We investigate 396 scenarios(problem instances) corresponding to 396 combinations of the warehouselength (T), the number of aisles (M), and the pick density (D). In all thesescenarios, the A, B, C parameters are respectively set to fixed values of 2.50,1.25, and 1.25 (m).

One fundamental parameter is the pick density (D), which is the averagenumber of items per main aisle. In each scenario, the total number of itemsto be picked is calculated as the multiplication of the pick density (D) withthe number of main aisles (M). The 11 pick density values listed above areused for calculating the order sizes during the estimation of average routelength for each scenario. Thus the 11 order sizes used in the experimentsare 0.1M, 0.5M, 1.0M, 1.5M, 2.0M, 2.5M, 3.0M, 3.5M, 4.0M, 4.5M, and 5.0M.

The parameter values in our study are selected such that we can extendthe experiments of Vaughan and Petersen (1999). Compared to the 126scenarios (combinations of T, M, and D) in their study, we consider 396scenarios. In addition, our parameters take values over broader ranges, wecalculate average travel distances over a greater number of instances (1000orders as opposed to 100 instances), and we consider Case 2 besides Case 1.

Table 5.1 Experimental Design

Factor Number of Values Values

396scen

arios Length of

main aisles (T) (m)6 30, 60, 90, 120, 150, 180

Number ofmain aisles (M)

6 5, 10, 15, 20, 25, 30

Pick density (D)(items/aisle)

11 0.1, 0.5, 1.0, 1.5, 2.0, 2.5,3.0, 3.5, 4.0, 4.5, 5.0

Number of cross aisles (N) 1 (for Case 0) 08 (for Case 1) 1, 2, 3, 4, 5, 6, 7, 83 (for Case 2) 1, 2, 3

(A, B, C) (m) 1 (2.50, 1.25, 1.25)

Note: A¼width of a cross aisle; B¼width of a main aisle; C¼width of a storageblock.



For each warehouse (T, M, N, A, B, C) and for each order size (D � M)we apply the following procedure:

Step 1: Generate a set of 1000 orders with D � M items each: Eachitem to be picked is assigned to a storage location by firstrandomly generating a main aisle number, and then, ran-domly generating the position within that main aisle on theinterval (0, T). The locations of the items to be picked in eachorder are assumed to be uniformly distributed across thewarehouse. This assumption can be encountered in relatedstudies (e.g., see Roodbergen and De Koster, 2001 AQ3).

Step 2: Apply RGSA.Step 2.a: Apply GSA for the generated set of orders and the given

warehouse. For each feasible configuration of storageblocks, the shortest-path dynamic programming algorithmof the V&P model is solved for each of the orders in the setof orders. Average travel distances in the set of orders isobtained and an initial best configuration of storage blocksthat provides the minimum average order-picking travel dis-tances is returned.

Step 2.b: Apply the remaining steps of RGSA. Given the initial solutionreturned by GSA, RGSA works on improving the Li valueswith the objective of minimizing average travel distance.

In Step 1 of the aforementioned procedure, the seed used to generate therandom numbers is always chosen the same. The result of the experimentsis a dataset with 396 rows and the following 17 columns: T, M, N, D, averagetravel length in Case 0, average travel length in Case 1 for N¼ 1, 2, . . . , 8(eight distinct columns), average travel length in Case 2 for N¼ 1, 2, 3 (threedistinct columns), area of the warehouse (for the scenario). We carry outour analysis in Section 7 using this dataset. The values in the dataset arecomputed through a heuristic algorithm (which is not optimal) and throughMonte Carlo simulation. Because we are using heuristic algorithms, fromnow on, the solution which will be referred to as the ‘‘best solution’’ isactually the incumbent solution, which is not necessarily optimal.

5.7 Analysis of Experimental Results

In this section, we analyze, through starfield visualizations, the results ofour computational experiments for the 396 scenarios. The first three figuresthat we discuss in this section (Figures 5.6 through 5.8) are referred to as‘‘colored scatter plots’’ in exploratory data analysis literature (Hoffman and



–3 –2 –1 0 1 2 3 4 5 6

6,25

5

a

Color = Pick density (0 to 5)Axis X = Percentage travel-time savingAxis Y = Percentage space saving

b

3,75

2,5

1,25

–1,25

–2,5

–3,75

–5

–6,25

–7,5

–8,75

0

6,25

5

3,75

2,5

1,25

–1,25

–2,5

–3,75

–5

–6,25

–7,5

–8,75

0

Figure 5.6 Savings with respect to the pick density (D).

–3 –2 –1 0 1 2 3 4 5 6

6,25

5

a

b

c

d

3,75

2,5

1,25

–1,25

–2,5

–3,75

–5

–6,25

–7,5

–8,75

0

6,25

5

3,75

2,5

1,25

–1,25

–2,5

–3,75

–5

–6,25

–7,5

–8,75

0

Color = T (0 to 180)Axis X = Percentage travel-time savingAxis Y = Percentage space saving

Figure 5.7 Savings with respect to the warehouse length (T).



Grinstein, 2002). According to this naming scheme, Figures 5.9 and 5.10are referred to as ‘‘jittered colored scatter plots,’’ and Figures 5.11 through5.13 are referred to as ‘‘colored 3-D scatter plots.’’ However, rather thanusing the terminology in exploratory data analysis, we refer to all theseplots as ‘‘starfield visualizations,’’ following the terminology in the field ofinformation visualization (Shneiderman, 1999). The starfield visualizationis an extended version of the scatter plot, with coloring, size, zooming,and filtering.

In each of Figures 5.6 through 5.13, information regarding which param-eter is mapped to which attribute of the plot/glyphs is displayed belowthe plot. For example, in Figure 5.6, D (pick density) values are mapped tocolor of the glyphs; percentage travel-time savings (in Case 2 comparedto Case 1) are mapped to the X-axis; and percentage space savings (in Case2 compared to Case 1) are mapped to the Y-axis. The range of pick densityvalues is 0 to 5. Lighter colors represent larger values of the mapped param-eter, and darker colors represent smaller values of themapped parameter. Allthe mappings are linear. Rectangular frames, such as frames (a) and (b) in

–3 –2 –1 0 1 2 3 4 5 6

6,25

5

Color = M (0 to 30)Axis X = Percentage travel-time savingAxis Y = Percentage space saving

3,75

2,5

1,25

–1,25

–2,5

–3,75

–5

–6,25

–7,5

–8,75

0

6,25

5

3,75

2,5

1,25

–1,25

–2,5

–3,75

–5

–6,25

–7,5

–8,75

0

Figure 5.8 Savings with respect to the number of main aisles (M).



0M

30

25

20

15

10

5

ab

Color = Percentage travel-time saving (–3 to 6)Axis X = T; Axis Y = M

0

20 40 60 80 100 120 140 160 180

30

25

20

15

10

5

0T

Figure 5.9 Percentage travel-time savings with respect to the warehouse length(T) and number of main aisles (M).

0M

30

25

20

15

10

5

a b

Color = Percentage space saving (–8.75 to 6.25)Axis X = T; Axis Y = M

0

30

25

20

15

10

5

0

20 40 60 80 100 120 140 160 180

T

Figure 5.10 Percentage space savings with respect to the warehouse length (T)and number of main aisles (M).



Color = Percentage travel-time saving, N = 1Axis X = T; Axis Y = M

20305

4,5

4

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3

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0,5020 40M

T

D

a

b

60 80 100 120 140 160 180

1

2

5

4,5

4

3,5

3

2,5

1,5

0,5

0

1

2

40 60 80 100 120 140 160 180

Figure 5.11 Percentage travel-time savings with respect to the warehouse length(T), number of main aisles (M), and pick density (D) for N 5 1.

Color = Percentage travel-time saving, N = 5Axis X = T; Axis Y = M

20305

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40 60 80 100 120 140 160 180

Figure 5.12 Percentage travel-time savings with respect to the warehouse length(T), number of main aisles (M), and pick density (D) for N 5 5.



Figure 5.6, are drawn to highlight specific regions in the plots that exhibit theinteresting properties.

5.7.1 Savings in Case 2 Compared to Case 1

Figures 5.6 through 5.10 illustrate the percentage savings gained in layoutswith unequally spaced cross aisles (Case 2) compared to layouts withequally spaced cross aisles (Case 1). In Figures 5.6 through 5.8, eachscenario is represented by a glyph (data point). The percentage travel-time (distance) savings in Case 2 compared to Case 1 are mapped to theX-axis, the percentage space savings are mapped to the Y-axis, and variousparameters (D, T, and M) are mapped to colors of the glyphs. In Figures 5.9and 5.10, each scenario is again represented by a glyph, but this time theT values are mapped to the X-axis, the M values are mapped to the Y-axis,and the percentage savings (in travel time and in warehouse space) aremapped to colors of the glyphs.

Color = “Best” no of equally spaced cross aisles (1 to 7)Axis X = T; Axis Y = M

2030 D5

4,5

4

3,5

3

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0,5

020 40M

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60 80 100 120 140 160 180

1

2

5

4,5

4

3,5

3

2,5

1,5

0,5

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1

2

40 60 80 100 120 140 160 180

Figure 5.13 The best number of cross aisles (N) with respect to the warehouselength (T), number of main aisles (M), and pick density (D).



In Figure 5.6, frame (a) shows that the scenarios with large per-centage travel-time savings are all characterized by high pick densities(large D values). Frame (b) shows that under certain scenarios, the percent-age travel-time savings are obtained only at the cost of big losses inwarehouse space (negative percentage space-saving values on the Y-axis).

In Figure 5.7, frame (a) shows that the scenarios with the largest per-centage travel-time savings are for warehouses that have medium T values.Frames (b) and (c) show that scenarios which benefit from Case 2 withrespect to percentage space savings are all characterized by large lengthvalues (light colors). However, in these scenarios there may be savings aswell as losses in percentage travel time, as can be seen in Frames (a) and(b), respectively. Frame (d), on the other hand, shows that in instances withshorter warehouses (glyphs with darker colors) the best number of crossaisles for Case 2 is more than the best number of cross aisles for Case 1, andthis can result in large percentage losses in warehouse space. Theseinstances are all characterized by high pick densities in from frame (b) ofFigure 5.6.

In Figure 5.8, the frame shows that the scenarios in which Case 2 resultsin significant space savings, but also the travel-time losses (negative valueson the X-axis) are all characterized by large M values (light tones of gray).From Figures 5.6 and 5.7, we remember that these are also scenarios withlow pick densities and largest T values.

The maximum percentage travel-time saving (X value of the rightmostglyph) obtained in the 396 scenarios is 5.28 percent. This result strikinglysuggests that unequally spaced cross aisles (Case 2) bring little additionalsavings in comparison to equally spaced cross aisles (Case 1). In our study,finding the best number and best positions of unequally spaced cross aislesrequired implementation of a nontrivial algorithm and allocation of signifi-cant running times for the computations (approximately ten days in total,most of it for computing the solutions for N¼ 3 in Case 2). Thus, we canconclude that warehouse planners are better off establishing rectangularwarehouses with equally spaced cross aisles instead of unequally spacedcross aisles. These results provide answers to the first research questionposed in Section 5.1.

Figures 5.9 and 5.10 allow the analysis of savings with respect to T andM, which are mapped to the X- and Y-axes, respectively. Because there are11 scenarios for each (T, M) pair, jittering is applied to a certain extent toavoid occlusion. So, in these two figures, the glyphs which are clusteredtogether have the same T and M values, but differ in their pick densities(D values).

In Figure 5.9, frame (a) shows that for small values of T, percentagetravel-time savings in Case 2 are higher in general (lighter glyphs). For large



warehouses, which are highlighted by frame (b), there are less savings, oreven losses in Case 2. This may be due to the fact that our algorithm findsthe best positions in Case 2 is run only for N¼ 1, 2, and 3. We thus believethat development of efficient algorithms for solving Case 2 for larger valuesof N is critical.

In Figure 5.10, frame (a) shows that shorter warehouses can incur bigspace losses in Case 2, because for the associated scenarios, the bestnumber of cross aisles required in Case 2 (to minimize travel time) istypically more than the number of cross aisles required in Case 1. Frame(b) shows, as expected, that there are space savings in Case 2 compared toCase 1, because N� 3 in Case 2 and N� 8 in Case 1. In general, through ourobservations in Section 7.1, we can conclude that it is sufficient to focusonly on Case 1, which we continue to analyze in Sections 5.7.2 and 5.7.3.

5.7.2 Impact of T, M, and D in Case 1

Figures 5.11 and 5.12 show the change in percentage travel-time savings inCase 1 (for N¼ 1 and N¼ 5) compared to Case 0, under the 396 combin-ations of T, M, and D (which are mapped to X-, Y-, and Z-axes, respect-ively). Percentage travel-time saving in each scenario is mapped to color ofthe related glyph. As shown in frame (a) of both figures, the largest travel-time savings are realized for pick densities (D values) between 0.5 and 2.5.This observation is consistent with earlier findings of Vaughan and Petersen(1999), who state that ‘‘the greatest cross aisle benefit occurs at pick dens-ities in the range 0.6 to 1.0 units/aisle,’’ which answers the second researchquestion posed in Section 5.1.

In both Figures 5.11 and 5.12, the impact of M is observed to benegligible, except for D¼ 0.1 (warehouses with very small orders). Thisconclusion is reached by observing that the colors of the glyphs do notchange significantly along the Y-axis, except for D¼ 0.1. Meanwhile, asshown in frame (b) of both figures, shorter warehouses (with smallT values) are most sensitive to changes in D.

Vaughan and Petersen (1999) also state that the ‘‘addition of cross aislesgenerally decreases the picking travel distance on average, with traveldistances frequently in the range 70 percent–80 percent, or even less, ofthat associated with the no cross aisles N¼ 0 layout,’’ quantifying thesavings that can be obtained. This is another significant finding of ourstudy. The percentage travel-time savings in Figures 5.11 and 5.12 takevalues between 0 and 35.30. That is, we observe travel-time savings of upto 35.30 percent by adding cross aisles, confirming the earlier findings ofthe authors.



5.7.3 Best Number of Cross Aisles in Case 1

Figure 5.13 shows the change in the best number of cross aisles in Case 1,which is mapped to color. The parameters T, M, and D are again mapped toX-, Y-, and Z-axes, respectively. Because none of the scenarios has its bestN value equal to 8, the range of N values is between 1 and 7. Also, becausethe glyphs in the figure do not have the lightest tones of gray, we can observethat the best N values are seldom 7 or 6. A count in the experimental resultsshows that in 389 (all but 7) of the 396 scenarios, the best N value is less thanor equal to 5. The most frequently encountered N value is 4, with 129scenarios implying that N¼ 4 is the most desirable number of cross aisles.

We can conclude from Figure 5.13 that the main determinant of the bestN value is the warehouse length T, because there are big changes in the colortone as one goes from smaller to larger values of T. By judging from thepattern of change in color tone, one can conclude that M also has someimpact as well. At this point, we can address the third research questionstated in Section 1 in two parts, both of which are very relevant andimportant. What is the best number of cross aisles for known values of Tand M? And is there a best number of cross aisles, regardless of T and M?

The first question is very relevant, because in designing warehouselayouts, warehouse planners typically have a very limited knowledge onfuture D values, while they generally have a good judgment of which valuesT and M should take. The answer to the first question is given in Table 5.2,

Table 5.2 The Best Number of Cross Aisles for Different (T, M) Pairs,Accompanied with the Maximum Travel-Time Losses that Onecan Encounter under Any D

T ¼ 30 T ¼ 60 T ¼ 90 T ¼ 120 T ¼ 150 T ¼ 180

M¼ 5 1 2 2 3 4 40.52a 1.32a 0.51a 0.30a 0.59a 0.37a

M¼ 10 1 2 3 3 4 41.86a 0.32a 0.23a 0.66a 0.58a 0.38a

M¼ 15 1 2 3 4 4 52.40a 0.86a 0.22a 0.10a 0.35a 0.21a

M¼ 20 2 3 3 4 4 52.55a 0.81a 0.44a 0.10a 0.31a 0.10a

M¼ 25 2 3 3 4 5 52.43a 0.74a 0.42a 0.18a 0.31a 0.12a

M¼ 30 2 3 3 4 5 52.45a 0.74a 0.54a 0.16a 0.27a 0.15a

a Percentage values.



which displays the best number of cross aisles for different (T, M) pairs,accompanied with the maximum travel-time losses that one can encounterunder any D values. For example, for (T, M)¼ (90, 10), the planner shouldestablish three cross aisles. Among the 11D values tested in our experimentsfor this (T, M) combination, establishing a different number of cross aislesresulted in at most 0.23 percent savings compared to three cross aisles.

The second question is very relevant as well, because many warehouseplanners would prefer to learn and always remember a single number,rather than having to refer to Table 5.2 in this chapter. Thus, the questionis which number of cross aisles to recommend to a warehouse plannerregardless of T, M, or D values. The answer to this question is simply three.A warehouse planner can always build warehouses with three equallyspaced cross aisles without compromising a significant loss in travel time,in particular, when compared to warehouses with different numbers ofcross aisles.

Of course, this result is valid if the A, B, C values are close to the valuesin Table 5.1, and the warehouse operates under the assumptions of ourmodel, including the usage of the aisle-by-aisle policy for routing. For thescenarios that we analyze in our experiments, the worst travel-time loss forthree cross aisles is 6.26 percent (Figure 5.14), and the worst warehousespace (area) loss is 15.38 percent (Figure 5.15). Figures 5.14 and 5.15 showthe worst average losses for other values of N.

If the warehouse space (area) is the most critical resource, then awarehouse planner can establish two cross aisles, instead of three. In ourexperiments, having two cross aisles resulted in at most 16.68 percent lossin travel time (Figure 5.14), and 7.69 percent loss in warehouse space(Figure 5.15). Thus, layouts with three (equally spaced) cross aisles are

60

50

40

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0

N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 N = 8N = 1

6.26 8.83

24.5530.11

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54.56

Figure 5.14 The maximum percentage gap—over all scenarios—between thetravel time under N and the best travel time for that scenario.



robust in terms of travel time, and layouts with two cross aisles are robust interms of warehouse space.

5.8 Conclusions and Future Work

In this chapter, we presented a detailed discussion of the impact of crossaisles on a rectangular warehouse. We analyzed both equally spaced andunequally spaced cross aisles, which we referred to as Case 1 and Case 2,respectively. For Case 1 we utilized the dynamic programming algorithmpresented in Vaughan and Petersen (1999) to determine the optimal orderpicking routes under aisle-by-aisle policy. For Case 2, we made modifica-tions to the Vaughan and Petersen (1999) model, including the change offormulas for certain parameters and introduction of new expressions beforerunning the dynamic programming algorithm. We computed the averagetravel times using Monte Carlo simulation for 396 distinct scenarios, whichcorrespond to 396 different warehouse and demand combinations. Ourprimary findings are:

& It is more desirable to establish only equally spaced cross aisles thanto establish unequally spaced cross aisles.

& Establishing (equally spaced) cross aisles can bring significanttravel-time savings and should definitely be considered. Weobtained savings up to 35.30 percent in our experiments. Biggesttravel-time savings are realized for pick densities between 0.5 and2.5 (D2 [0.5, 2.5]).

50

60

40

30

20

10

0N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 N = 8

15.38 15.38

7.69

23.08

30.77

38.46

46.15

53.85

Figure 5.15 The maximum percentage gap—over all scenarios—between thewarehouse space under N and the warehouse space for the best N for thatscenario.



& Given the length of main aisles and the number of main aisles(T and M), warehouse planners can refer to Table 5.2 in this chapterto determine the best number of (equally spaced) cross aisles. Ifone does not wish to refer to this table, but wishes to learn andremember a single value for the best number of cross aisles, wepropose the value of three.

There are several directions for future research relating to our study. We listtwo of those:

& It is necessary to design faster and better algorithms to identify thebest storage-block lengths (Li) in Case 2, and to validate further ourfirst conclusion.

& It is important to test the robustness of our conclusions under otherdemand patterns and different routing heuristics.

Our study contributes to the research and practice of warehouse planning/facility logistics by providing actionable insights regarding cross aisles.There is a great potential for research that attempts to solve warehousingproblems that require taking interdependent decisions at different timehorizons; for example, at both strategic and operational levels. Finally, wesuggest the adoption of data analysis techniques from the field of informa-tion visualization for discovering knowledge hidden in experimental andempirical data related to warehouse planning.

Acknowledgments

The authors thank Kemal Kilic at Sabanci University for his extensive helpin reviewing the M.S. thesis of Bilge Incel (Kucuk), which in time turnedinto this chapter. The authors also thank S. Ilker Birbil for reviewing thefinal draft of the paper and suggesting many improvements.

References

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De Koster, R. and Van der Poort, E. 1998. ‘‘Routing order pickers in a warehouse:A comparison between optimal and heuristic solutions,’’ IIE Transactions,30, 469–480.



Hoffman, P.E. and Grinstein, G.G. 2002. ‘‘A survey of visualiations for high-dimensional data mining,’’ in U. Fayyad, G. Grinstein, and A. Wierse(Eds.), Information Visualization in Data Mining and Knowledge Discov-ery. Morgan Kaufmann, San Fransisco, CA.

Frazelle, E.H. 2001.World-Class Warehousing andMaterial Handling. McGraw-Hill,New York.

Frazelle, E.H. and Apple, J.M. 1994. ‘‘Warehouse operations,’’ in J.A. Tompkinsand D. Harmelink (Eds.), The Distribution Management Handbook.McGraw-Hill, New York.

Keim, D.A. 2002. ‘‘Information visualization and visual data mining,’’ IEEE Trans-actions on Visualization and Computer Graphics, 8(1), 1–8.

Petersen, C.G. 1997. ‘‘An evaluation of order picking routing policies,’’ Inter-national Journal of Operations and Production Management, 17, 1098–1111.

Ratliff, H.D. and Rosenthal, A.S. 1983. ‘‘Order Picking in a rectangular warehouse:A solvable case of the traveling salesman problem,’’ Operations Research,31(3), 507–521.

Roodbergen, K.J. and De Koster, R. 2001a. ‘‘Routing order pickers in a warehousewith a middle aisle,’’ European Journal of Operational Research, 133, 32–33.

Roodbergen, K.J. and De Koster, R. 2001b. ‘‘Routing methods for warehouses withmultiple cross aisles,’’ International Journal of Production Research, 39,1865–1883.

Rouwenhorst, B., Reuter B., Stockrahm, V., Van Houtum, G.J., Mantel, R.J., andZijm, W.H.M. 2000. ‘‘Warehouse design and control: Framework and litera-ture review,’’ European Journal of Operational Research, 122, 515–533.

Sharp, P.G. 2000. ‘‘Warehouse management,’’ in G. Salvendy (Ed.), Handbook ofIndustrial Engineering. John Wiley, New York.

Shneiderman, B. 1999. ‘‘Dynamic queries, starfield displays, and the path to Spot-fire.’’ Available at http://www.cs.umd.edu/hcil/spotfire/. Retrieved onNovember 2006.

Spence, R. 2001. Information Visualization. ACM Press, Essex, UK.Van den Berg, J.P. and Zijm, W.H.M. 1999. ‘‘Models for warehouse management:

Classification and examples,’’ International Journal of Production Econo-mics, 59, 519–528.

Vaughan, T.S. and Petersen, C.G. 1999. ‘‘The effect of warehouse cross aisles onorder picking efficiency,’’ International Journal of Production Research, 37,881–897.



Appendix

GRID SEARCH ALGORITHM (GSA)This algorithm returns bestL, the best block lengths among tested lay-

outs, for a given N. The length of the gridForL array is (N þ 1) and indicatesthe number of storage blocks. Moreover, gridsForL[i] records the number ofgrids that constitute the length of the ith storage block. If the summation ofthe elements of gridforL array is equal to noOfGrids value, a feasiblestorage-block length combination is obtained. When noOfGrids¼ 20 andN¼ 2, for example, then some of the feasible storage-block lengths (L1, L2,L3) would be (1, 12, 7), (11, 4, 5), having the summation of L values equal tonoOfGrids¼ 20.

GSA generates feasible configurations of storage blocks systematicallyand returns the travel distances by solving the modified model that wepresent for a uniformly distributed set of orders. Average of the traveldistances for the order set is taken and the initial best configuration ofstorage blocks enabling the minimum average order-picking travel dis-tances is labeled as bestL.

This algorithm generates a greater number of feasible storage-blocklength alternatives as the number of grids is increased. This results insmaller unit length (G¼ T/noOfGrids). However, the more the number offeasible solution gets, the more will be the computational effort. Weobserved in our experiments that for the warehouse and order settingsdescribed in the next section, noOfGrids¼ 7 is computationally prohibitive(18 days running time including the cases where N¼ 4), and noOfGrids isselected as 7.

P¼ {1,. . . . , N þ 1}, O¼ {1,. . . . , u}

GRID_SEARCH_ALGORITHM (warehouse, orders, noOfGrids)G¼T/noOfGridsfor each gridsForL, /* s.t. gridsForL[i] � noOfGrids, 8i */sumOfGrids¼P

i2�gridsForL[i]


123

if(sumOfGrids¼ ¼ noOfGrids&&ARRAY_CONTAINS_NOZERO(gridsForL))

tempL[i]¼ gridsForL[i] * G, 8i 2 PTempWarehouse.setL(tempL)

orders[o].setWarehouse(tempWarehouse), 8o 2 OtempSimulationStatistics¼CALCULATE_SIMULATION_STATISTICS(orders)

tempTravelDistance¼ tempSimulationStatistics.getAverage( )if (tempTravelDistance < bestTravelDistance)

bestL¼ tempLbestTravelDistance¼ tempTravelDistance

return bestL

CALCULATE_SIMULATION_STATISTICS (orders)

travelDistance[o]¼ getOptimalTravelDistance(orders[o]), 8o 2 Oreturn statistics for travelDistance data

REFINED GRID SEARCH ALGORITHM (RGSA)This algorithm starts with the result of the GSA as the initial solution, and

applies changes in little unit lengths (G) to the initial best configuration ofstorage blocks (initialBestL). In this method, first a range is defined. Half ofthis range is subtracted from each storage-space length and smaller unitlengths (gridsForL[i] � G) are added to each storage-space length. Thetravel distance for the new configuration tempL is calculated for the givenorder set (orders) and compared with the best result obtained until thattime. After trying all feasible configurations of the gridsForL for the sameinitial solution and calculating the travel distance for the new storage-blocklengths, tempL resulting in the shortest travel distance is assigned as the bestconfiguration of cross aisles, bestL. Then the range is updated by dividingwith the number of grids (noOfGrids). Half of this range is subtracted fromeach storage-space length and smaller unit lengths (gridsForL[i] � G) areadded to obtain new feasible storage-block lengths (tempL) and traveldistance implied by the updated tempL is calculated for the given orderset (orders). The refined grid search is continued until the range declines toa length, which is determined as the smallest range (resolution) to beconsidered. When the range becomes as small as the resolution, the refinedgrid search is terminated and the improved configuration of storage-blocklengths is assigned as the best configuration of storage-block lengths (bestL)for the given warehouse and order set.

Any element of gridsForL can be at most (N þ 1) � noOfGrids, becausein the RGSA for each storage block, half of the range is subtracted and thelength gridsForL[i] � G is added, for instance:



From the above equations it is clearly seen that summation of the grid-sForL’s elements has to be (N þ 1) � noOfGrids. Therefore, an element ofgridsForL is allowed to be (N þ 1) � noOfGrids at most.

The search algorithms result in the best storage-block lengths that givethe minimum order-picking travel distance for a problem instance (T, M, N,A, B, C, D) among the tested configurations.

REFINED GRID SEARCH ALGORITHM(warehouse, orders, noOfGrids,resolution)

initialBestL¼GRIDSEARCHALGORITHM(warehouse,orders, noOfGrids)range¼ T/noOfGridsiterationNo¼ 0continueFlag¼ truewhile (continueFlag)iterationNoþþif (iterationNo > 1) // if not the first iterationrange¼ (range/noOfGrids)/2

G¼ (range/noOfGrids)/2for each gridsForL /� gridsForL[i] � (N þ 1) � noOfGrids �/sumOfGrids¼P

i2�gridsForL[i]

if (sumOfGrids¼ ¼ (N þ 1)noOfGrids&&ARRAY_CONTAINS_NOZERO

(gridsForL))

tempL[i]¼ initialBestL[i] � (range/2) þ gridsForL[i]*G, 8i 2 PtempWarehouse.setL(tempL)orders[o].setWarehouse(tempWaarehouse), 8o 2 OtempSimulationStatistics¼CALCULATE_SIMULATION_STATISTICS

(orders)

tempTravelDistance¼ tempSimulationStatistics.getAverage( )if (tempTravelDistance < bestTravelDistance)

bestL¼ tempLbestTravelDistance¼ tempTravelDistance

if (range<resolution)continueFlag¼ false

return bestL

AUTHOR QUERIES

[AQ1] Is the change correct?[AQ2] Roodbergen and De Koster 2001a or b?[AQ3] 2001a or b?



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